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%I #21 Feb 10 2018 22:00:07
%S 0,0,1,0,1,2,4,2,4,6,9,6,9,12,16,12,16,20,25,20,25,30,36,30,36,42,49,
%T 42,49,56,64,56,64,72,81,72,81,90,100,90,100,110,121,110,121,132,144,
%U 132,144,156,169,156,169,182,196,182,196,210,225,210,225,240
%N Sum of the smaller parts of the partitions of n into two distinct parts with the larger part even.
%C a(n+1) is the sum of the smaller parts in the partitions of n into two parts with the larger part odd. For example, a(11) = 9; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Three of these partitions have an odd number as their larger part, namely (9,1), (7,3) and (5,5). Adding the smaller parts of these partitions gives 1 + 3 + 5 = 9.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor((n-1)/2)} i * ((n-i+1) mod 2).
%F Conjectures from _Colin Barker_, Dec 06 2017: (Start)
%F G.f.: x^3*(1 - x + x^2 + x^3) / ((1 - x)^3*(1 + x)^2*(1 + x^2)^2).
%F a(n) = a(n-1) + 2*a(n-4) - 2*a(n-5) - a(n-8) + a(n-9) for n > 9.
%F (End)
%F a(n) = floor((n+1)/4)^2*(n mod 2)+(1+floor((n-2)/4))*floor((n-2)/4)*((n+1) mod 2). - _Wesley Ivan Hurt_, Dec 08 2017
%e a(10) = 6; the partitions of 10 into two parts are (9,1), (8,2), (7,3), (6,4) and (5,5). Two of these partitions have an even number as their larger part, namely (8,2) and (6,4). Adding the smaller parts of these partitions gives 2 + 4 = 6.
%t Table[Sum[i Mod[n - i + 1, 2], {i, Floor[(n - 1)/2]}], {n, 80}]
%o (PARI) a(n) = sum(i=1, floor((n-1)/2), i*lift(Mod(n-i+1, 2))) \\ _Iain Fox_, Dec 06 2017
%Y Cf. A295287, A295293.
%K nonn,easy
%O 1,6
%A _Wesley Ivan Hurt_, Dec 05 2017
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